Math CentralQuandaries & Queries


Question from M:

Given any 3 parallel lines on a plane, how to construct an equilateral triangle with each vertex on each line?

What works here is a strategy that is often helpful in construction problems: work backwards. Start with an equilateral triangle ABC and take an arbitrary line, call it a, through vertex A, and draw lines b and c parallel to a through B and C. Try rotating your picture through 60 degrees about A (so that B rotates B to C), and observe what happens to to line b during this rotation. (Warning: when working forwards from the three given parallel lines, there are generally two solution triangles; you can rotate line b clockwise and counterclockwise about the point A taken arbitrarily on line a.)


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